Method, apparatus and design procedure for controlling multi-input, multi-output (MIMO) parameter dependent systems using feedback LTI&#39;zation

ABSTRACT

A method and apparatus are provided for controlling a dynamic device having multi-inputs and operating in an environment having multiple operating parameters. A method of designing flight control laws using multi-input, multi-output feedback LTI&#39;zation is also provided. The method includes steps of: (i) determining coordinates for flight vehicle equations of motion; (ii) transforming the coordinates for the flight vehicle equations of motion into a multi-input linear time invariant system; (iii) establishing control laws yielding the transformed equations of motion LTI; (iv) adjusting the control laws to obtain a desired closed loop behavior for the controlled system; and (v) converting the transformed coordinates control laws to physical coordinates.

This application is a continuation of application Ser. No. 09/580,587,filed May 30, 2000, now U.S. Pat. No. 6,539,290, incorporated herein byreference.

BACKGROUND OF THE INVENTION

I. Field of the Invention

The present invention relates to a method of designing control laws(e.g., flight control laws in an airplane) by applying a techniquecalled Multi-Input, Multi-Output (MIMO) feedback LTI'zation, which isapplicable to solving a feedback control design problem for a class ofnonlinear and linear parameter dependent (“LPD”) dynamic systems, alsoknown as linear parameter varying (“LPV”), with multiple inputs andmultiple outputs. Feedback LTI'zation combines a co-ordinatestransformation and a feedback control law, the results of which cancelsystem parameter dependent terms and yield the transformed space openloop system linear time invariant (LTI). The present invention furtherrelates to using multi-input feedback LTI'zation to solve the controldesign problem associated with control systems for LPD dynamic devices.In particular, the invention is applied to a feedback control system forcontrolling a parameter dependent dynamic device (e.g., an airplane)with multiple control inputs.

II. Background

Design techniques used for solving feedback control design problems canbe divided into several classes. For example, two broad classes are (1)Linear Time Invariant systems (herein after referred to as “LTI”) and(2) nonlinear systems. In the last four decades, LTI systems havereceived a great deal of attention resulting in many well-definedcontrol design techniques. See, e.g., Maciejowski, J. M., MultivariableFeedback Design, 1989, Addison-Wesley and Reid, J. G., Linear SystemFundamentals, 1983, McGraw-Hill, each incorporated herein by reference.Nonlinear systems have, in contrast, received far less attention.Consequently, a smaller set of techniques has been developed for use infeedback control system design for nonlinear systems or linear parameterdependent systems. As a result, control law design for nonlinear systemscan be an arduous task. Typically, control laws consist of a pluralityof equations used to control a dynamic device in a desirable andpredictable manner. Previously, designing control laws for LPD systemsusing quasi-static LTI design techniques could require an enormousamount of effort, often entailing weeks, if not months, of time tocomplete a single full envelope design. For example, when designing aflight control law, designers must predict and then design the controllaw to accommodate a multitude (often thousands) of operating pointswithin the flight envelope (i.e., the operating or performance limitsfor an aircraft).

Feedback Linearization (reference may be had to Isidori, A., NonlinearControl Systems, 2nd Edition, 1989, Springer-Verlag, herein incorporatedby reference), is applicable to control design for a broad class ofnonlinear systems, but does not explicitly accommodate system parameterchanges at arbitrary rates. Feedback LTI'zation, a technique used forrendering a control system model linear time invariant, for single inputsystems is outlined in the Ph.D. thesis of the inventor, Dr. David W.Vos, “Non-linear Control Of An Autonomous Unicycle Robot; PracticalIssues,” Massachusetts Institute of Technology, 1992, incorporatedherein by reference. This thesis extends Feedback Linearization toexplicitly accommodate fast parameter variations. However, the Ph.D.thesis does not give generally applicable solutions or algorithms forapplying feedback LTI'zation to either single input or multi-inputparameter dependent dynamic systems. U.S. Pat. No. 5,615,119 (hereinincorporated by reference, and hereafter the “'119”patent) addressedthis problem, albeit in the context of failure detection filter design.In particular, the '119 patent describes a fault tolerant control systemincluding (i) a coordinate transforming diffeomorphism and (ii) afeedback control law, which produces a control system model that islinear time invariant (a feedback control law which renders a controlsystem model linear time invariant is hereinafter termed “a feedbackLTI'ing control law”).

The '119 patent encompasses fault detection and isolation and controllaw reconfiguration by transforming various actuator and sensor signalsinto a linear time invariant coordinate system within which an LTIfailure detection filter can be executed, to thus provide a capabilityfor failure detection and isolation for dynamic systems whose parametersvary over time. That is, a detection filter may be implemented in aso-called Z-space in which the system may be represented as linear timeinvariant and is independent of the dynamic system parameters.

What is needed, however, is the further extension of the feedbackLTIzation control law principals in the '119 patent to multi-inputparameter dependent systems. Furthermore, control system designers havelong experienced a need for a fast and efficient method of designingcontrol laws relating to parameter dependent nonlinear systems. Anefficient method of control law design is therefore needed. Similarly,there is also a need for a control system aimed at controlling such adynamic device with multiple control inputs.

SUMMARY OF THE INVENTION

The present invention specifically solves a Multi-Input feedbackLTI'zation problem, and shows a method for feedback control law designfor a parameter dependent dynamic device (e.g., an airplane) class ofsystems. Additionally, the present invention provides a control systemfor controlling a parameter dependent dynamic device with multipleinputs. The present invention is also applicable to the methods andsystems discussed in the above-mentioned '119 patent (i.e., for failuredetection system design in the multi-input case). As a result of theconcepts of this invention, control system designers may now shave weeksor months off of their design time.

According to one aspect of the invention, an automatic control systemfor controlling a dynamic device is provided. The device includessensors and control laws stored in a memory. The control system includesa receiving means for receiving status signals (measuring the statevector) and current external condition signals (measuring parametervalues) from the sensors, and for receiving reference signals. Alsoincluded is processing structure for: (i) selecting and applying gainschedules to update the control laws, wherein the gain schedulescorrespond to the current external conditions signals (parameter values)and are generated in a multi-input linear time invariant coordinatessystem; (ii) determining parameter rates of change and applying theparameter rates of change to update the control laws; (iii) applyingdevice status signal feedback to update the control laws; and (iv)controlling the device based on the updated control laws.

According to another aspect of the invention, a method for designingflight control laws using multi-input parameter dependent feedback isprovided. The method includes the following steps: (i) determining acoordinates system for flight vehicle equations of motion; (ii)transforming the coordinates system for the flight vehicle equations ofmotion into a multi-input linear time invariant system; (iii)establishing control criteria yielding the transformed coordinatesequations of motion LTI; (iv) adjusting the control criteria to obtain adesired closed loop behavior for the controlled system; and (v)converting the transformed coordinates control laws to physicalcoordinates.

According to still another aspect of the invention, a method ofcontrolling a dynamic device is provided. The device includingactuators, sensors and control laws stored in a memory. The methodincludes the following steps: (i) transforming device characteristicsinto a multi-input linear time invariant system; (ii) selecting andapplying physical gain schedules to the control laws, the gain schedulescorresponding to the current external condition signals; (iii)determining and applying parameter rates of change to update the controllaws; (iv) applying device status signal feedback to update the controllaws; (v) converting the transformed coordinates control laws tophysical coordinates; and (vi) controlling the device based on theupdated control laws.

Specific computer executable software stored on a computer or processorreadable medium is also another aspect of the present invention. Thissoftware code for developing control laws for dynamic devices includes:(i) code to transform device characteristics into a multi-input lineartime invariant system; (ii) code to establish control criteria yieldingthe transformed coordinates equations of motion LTI; (iii) code todefine a design point in the multi-input linear time invariant system;(iv) code to adjust the transformations to correspond with the designpoint; and (v) code to develop a physical coordinates control lawcorresponding to the adjusted transformations; and (vi) code to applyreverse transformations to cover the full design envelope.

In yet another aspect of the present invention, a multi-input parameterdependent control system for controlling an aircraft is provided. Thesystem includes receiving means for receiving aircraft status signalsand for receiving current external condition signals. A memory having atleast one region for storing computer executable code is also included.A processor for executing the program code is provided, wherein theprogram code includes code to: (i) transform the aircraftcharacteristics into a multi-input linear time invariant system; (ii)select and apply gain schedules to flight control laws, the gainschedules corresponding to the current external condition signals; (iii)determine parameter rates of change, and to apply the parameter rates ofchange to the flight control laws; (iv) apply feedback from the aircraftstatus signals to the flight control laws; (v) convert the transformedcoordinates control laws to physical coordinates; and (vi) control theaircraft based on the updated flight control laws.

BRIEF DESCRIPTION OF THE DRAWINGS

The present invention will be more readily understood from a detaileddescription of the preferred embodiments taken in conjunction with thefollowing figures.

FIG. 1 is a perspective view of an aircraft incorporating an automaticcontrol system of the present invention.

FIG. 2 is a functional block diagram describing an algorithm accordingto the present invention.

FIG. 3 is a flowchart showing a software flow carried out in the flightcontrol computer of FIG. 1.

FIG. 4 is a block diagram of the flight control computer, sensors, andactuators, according to the FIG. 1 embodiments.

FIGS. 5 a and 5 b are overlaid discrete time step response plotsaccording to the present invention.

FIGS. 5 c and 5 d are overlaid Bode magnitude plots according to thepresent invention.

FIG. 6 is an S-plane root loci plot for the closed and open loop lateraldynamics according to the present invention.

FIGS. 7-18 are 3-D plots of auto pilot gains vs. air density and dynamicpressure according to the present invention.

FIGS. 19-30 are numerical gain lookup tables according to the presentinvention.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Described herein is a technique called Multi-Input Feedback (“FBK”)LTI'zation, which is applicable to solving feedback control designproblems for a class of nonlinear and linear parameter dependent systemswith multiple inputs such as actuator commands. This techniqueaccommodates arbitrary changes and rates of change of system parameters,such as air density and dynamic pressure. As will be appreciated by oneof ordinary skill in the art, a subset of nonlinear systems, namelylinear parameter dependent (herein after referred to as “LPD”) systems,is one method of modeling real world dynamic systems. Control design forsuch systems is traditionally achieved using LTI (linear time invariant)design techniques at a number of fixed parameter values (operatingconditions), where at each operating condition the system's equations ofmotion become LTI. Gain scheduling by curve fitting between these designpoints, is then used to vary the gains as the operating conditions vary.Feedback LTI'zation gives a simple and fast method for full envelopecontrol design, covering any parameter value. In addition, the resultinggain schedules (as discussed below) are an automatic product of thisdesign process, and the closed loop system can be shown to be stable forthe full parameter envelope and for arbitrary rates of variation of thesystem parameters throughout the operating envelope, using these gainschedules and the feedback LTI'ing control law.

The process of applying Feedback LTI'zation to facilitate designingcontrol laws involves several steps, including: (i) transforming thecoordinates of equations of motion of a dynamic device (e.g., anairplane) into a so-called z-space; (ii) defining a control law whichyields the transformed coordinates equations of motion linear timeinvariant (LTI); and (iii) applying LTI design techniques to thetransformed coordinates mathematical model to yield a desired closedloop behavior for the controlled system, all as discussed below. Thethird step is achieved by (a) designing the feedback gains in physicalcoordinates at a selected operating condition; (b) using the coordinatestransformations to map these gains into z-space; and (c) reverse mappingvia the coordinates transformations and Feedback LTI'ing control laws todetermine physical coordinates control laws for operating conditionsother than at the design conditions.

One aspect of the present invention will be described with respect to anaircraft automatic flight control system for maintaining desiredhandling qualities and dynamic performance of the aircraft. However, thepresent invention is also applicable to other dynamic devices, such asvehicles including automobiles, trains, and robots; and to other dynamicsystems requiring monitoring and control. Furthermore, the presentinvention encompasses a design method and system for designing controllaws by, for example, defining a point in z-space, and then updating asystem transformation to generate control laws in x-space (i.e., inphysical coordinates).

FIG. 1 is a perspective view of an aircraft 1 having flight controlsurfaces such as ailerons 101, elevator 102, and rudder 103. Forexample, aircraft 1 is a Perseus 004 unmanned aircraft operated byAurora Flight Science, Inc. of Manassas, Va. Each flight control surfacehas an actuator (not shown in FIG. 1) for controlling the correspondingsurface to achieve controlled flight. Of course, other flight controlactuators may be provided such as throttle, propeller pitch, fuelmixture, trim, brake, cowl flap, etc.

The actuators described above are controlled by a flight controlcomputer 104 which outputs actuator control signals in accordance withone or more flight control algorithms (hereinafter termed “flightcontrol laws”) in order to achieve controlled flight. As expected, theflight control computer 104 has at least one processor for executing theflight control laws, and/or for processing control software oralgorithms for controlling flight. Also, the flight computer may have astorage device, such as Read-Only Memory (“ROM”), Random Access Memory(“RAM”), and/or other electronic memory circuits.

The flight control computer 104 receives as inputs sensor status signalsfrom the sensors disposed in sensor rack 105. Various aircraftperformance sensors disposed about the aircraft monitor and providesignals to the sensor rack 105, which in turn, provides the sensorsignals to the flight control computer 104. For example, providedaircraft sensors may include: an altimeter; an airspeed probe; avertical gyro for measuring roll and pitch attitudes; rate gyros formeasuring roll, pitch, and yaw angular rates; a magnetometer fordirectional information; alpha-beta air probes for measuring angle ofattack and sideslip angle; etc. As will be appreciated, if a roll ratesensor is not included in the sensor suite or rack 105, a roll ratesignal may be synthesized by using the same strategy as would be used ifan onboard roll rate sensor failed in flight. Meaning that the roll ratesignal is synthesized by taking the discrete derivative of the rollattitude (bank angle) signal. The manipulation (i.e., taking thediscrete derivative) of the bank angle signal may be carried out bysoftware running on the flight computer 104. Thus, using sensor statusinputs, control algorithms, and RAM look-up tables, the flight controlcomputer 104 generates actuator output commands to control the variousflight control surfaces to maintain stable flight.

FIG. 2 is a functional block diagram illustrating functional aspectsaccording to the present invention. FIG. 2 illustrates an algorithmaccording to the present invention including flight control laws for atwo-control input (e.g., rudder and aileron), as well as a two-parameter(e.g., air density and dynamic pressure) system. In particular, thisalgorithm controls the lateral dynamics of an unmanned aircraft (e.g.,the Perseus 004 aircraft). As will be appreciated by one of ordinaryskill in the art, a system using more than two inputs or more than twoparameters is a simple extension of the principles and equationsdiscussed below. Therefore, the invention is not limited to a case orsystem using only two inputs or two parameters. Instead, the presentinvention may accommodate any multiple input and multiple parametersystem.

As is known, a mathematical model of the aircraft (e.g., aparameter-dependent dynamic system) depicted in FIG. 1 may be written ina physical coordinate system (hereinafter called a coordinate system inx-space). In the case of an aircraft, a Cartesian axis system may haveone axis disposed along the fuselage toward the nose, one axis disposedalong the wing toward the right wing tip, and one axis disposed straightdown from the center of mass, perpendicular to the plane incorporatingthe first two axes. Measurements via sensors placed along or about theseaxes provide information regarding, for example, roll rate, bank angle,side slip, yaw rate, angle of attack, pitch rate, pitch attitude,airspeed, etc.

The '119 patent provides a description of the methodology for solvingthe problem of finding a state space transformation and feedback controllaw for a single input linear parameter dependent system. The solutionsto these problems yield linear parameter dependent (“LPD”; also referredto as linear parameter varying—“LPV”) coordinates transformations,which, when applied to system model equations of motion together withFeedback LTI'ing control laws, yield descriptions that are linear timeinvariant (LTI) in the transformed state space (z-space). As discussedin the '119 patent, a detection filter may be implemented in z-space inwhich the dynamic system (e.g., an aircraft) may be represented aslinear time invariant and, as such, is independent of the dynamic systemparameters. Basically, “non-stationary” aircraft flight dynamicsequations are transformed into “stationary” linear equations in ageneral and systematic fashion. In this context, “stationary” impliesthat the dynamic characteristics are not changing. As a result, a set ofconstant coefficient differential equations is generated in z-space formodeling the system. The combination of state space transformation andfeedback control law (this control law is called a “Feedback LTI'ingcontrol law”), which cancels all the parameter dependent terms, isreferred to as “Feedback LTI'zation”. By way of example, the solution asdescribed in U.S. Pat. No. '119 would be applicable to control of thelongitudinal aircraft dynamics problem, using a single input (e.g., theelevator).

The present invention details the further extension of FeedbackLTI'zation to accommodate a multi-input case which is particularlyrelevant to the lateral axis of a conventional aircraft, using a rudderand ailerons as actuators, for example. As such, the present inventionencompasses a control system for controlling dynamic devices and afailure detection system. This formulation is also relevant to highperformance aircraft, which may have multiple actuators in both thelateral and longitudinal axes.

In physical terms, the present problem is similar to the one faced in asingle input case; namely, a mathematical description of the vehicledynamics must be found, which does not change as the parameters of thevehicle change. In other words, it is desirable to rewrite a systemequations of motion such that the dynamic behavior of the system isalways the same and thus very predictable, regardless of what theoperating conditions or operating parameters are (e.g., the systemdynamics need to be expressed as linear time invariant (LTI) for anyparameter value or any rate of change of the parameter values).

This process of describing the equations of motion according to theabove requirements can be achieved through a combination of coordinatechange and feedback control laws. A coordinate transformation, ordiffeomorphism, is determined which transforms the physical coordinates(i.e., x-space) description of the aircraft dynamic mathematical modelto a new set of coordinates (i.e., z-space). A Feedback LTI'ing controllaw is defined to cancel all the z-space parameter dependent terms, sothat with this control law, the z-space system is then independent ofthe parameters. In fact, the behavior of the model in the transformedcoordinates is then that of a set of integrators, independent of vehicleoperating conditions. It is then possible to prescribe the desiredclosed loop behavior by means of LTI control design techniques, appliedin x-space and transformed to z-space, which is then also valid for anypoint in the operating envelope (e.g., sea level to 22 km above sealevel, and 20 m/s to 46.95 m/s indicated airspeed (IAS)), and forarbitrary values and rates of change of the parameters defining theoperating envelope. In this manner only a single, or at worst, a smallnumber of points in the operating envelope may be identified as designpoints. The diffeomorphism (transformation) has very specific dependenceon the parameter values. By evaluating the parameter values and thenevaluating the diffeomorphisms at these parameter values, the applicablegains are automatically defined appropriate for the current operatingconditions, and thus flight control laws may be obtained for use in thephysical coordinates (x-space).

As will be appreciated by those skilled in the art, the process ofdesigning control systems is fundamentally based on meeting requirementsfor achieving desired closed loop characteristics or behavior. Personsskilled in the art of LTI control design use well known techniques andfollow standard procedures to achieve this. In the multivariable controldomain for LTI systems, it is indeed possible to prescribe the desiredclosed loop dynamics of all the characteristic motions of the vehicle,and a known algorithm such as “pole-placement” can be used to determinethe gains that will deliver this. Other known techniques such as LQR(Linear Quadratic Regulator theory) can be used to achieve the samegoal.

As stated above, the coordinate change is mathematical, and allows asimple and easy mathematical treatment of the full envelope controldesign problem. The Feedback LTI'ing control law can also be describedmathematically, however, it is implemented physically and involves aspecific set of control algorithms. These control algorithms, combinedwith the coordinates change, result in the closed loop controlledphysical dynamic behavior that is repeatable and predictable at allregions of the flight envelope.

The problem of finding such a coordinates change (diffeomorphism) andcontrol law, is a central focus of both the feedback linearization andfeedback LTI'zation (for linear time varying parameters) problems.Essentially, the problem primarily requires solution of the coordinatestransformation matrix. After a solution is found, the rest of the designprocess follows as a matter of course. Clearly, this coordinate changeshould not result in any loss of information about the vehicle dynamicbehavior. In other words, the specific outputs combinations which ensurethat all the dynamic information is retained when observing the systembehavior from a different set of coordinates must be found. Finding thespecific outputs combinations has a direct relationship with transferfunctions, which describe how inputs reach outputs in a dynamic sense.That is, the solution is achieved when the new coordinates result in thefeature that all inputs will reach the outputs in a dynamic sense, andare not masked by internal system behavior. As will be appreciated, thecoordinates transformation should occur smoothly (e.g., without loss ofdata or without singularity) in both directions, i.e., from the model(z-space) description in one set of coordinates to the physicalcoordinates (x-space) description.

Once these output functions, or measurement directions, are known, itbecomes a fairly mechanical process to determine the coordinatestransformation and feedback control law, respectively. In fact, thepresent invention establishes that for LPD systems, the entire processof both finding the measurement direction, as well as the diffeomorphismand control laws, all become straightforward procedures, which can alsobe automated.

Solution of the Feedback LTI'zation Problem for LPD Multi-Input DynamicSystems

In order to define the diffeomorphism (Φ) and the Feedback LTI'ingcontrol law (ν), consider the affine multi-input parameter dependentsystem given by the equations: $\begin{matrix}{{\overset{.}{x} = {{f\left( {x,p} \right)} + {\sum\limits_{i = 1}^{m}{{g_{i}\left( {x,p} \right)}u_{i}}}}}\begin{matrix}{y_{1} = {h_{l}(x)}} \\\vdots \\{y_{m} = {h_{m}(x)}}\end{matrix}} & (1)\end{matrix}$where x∈R^(n); u, y∈R^(m); p∈R^(q) and where x is a state vector made upof state variables, such as roll rate, yaw rate, side slip, bank angle,etc., u_(i) is the i^(th) control input (e.g., rudder or aileron, etc.),and the output measurement directions, h_(i)(x), are to be determinedappropriately in order to define the state variable transformation. Thefunctions f(.) and g(.) are functions of both the state vector x as wellas the parameter vector p.

First, a multivariable definition of relative degree is needed, namely avector relative degree, which pertains to the number of zeros in thetransfer function from the input vector u to the output vector y. Thisdefinition is taken directly from Isidori, discussed above.

Definition

A multivariable system of the form (1) with m inputs and m outputs, hasvector relative degree {r₁, r₂, . . . , r_(m)} at a point x_(o) if thefollowing hold:

-   1) For any 1≦i,j≦m and for all k≦r_(i)−1,    L _(gi) L _(f) ^(k) h _(i)(x)=0  (2)    where the operator “L” is the Lie derivative.-   2) The m by m matrix A(x) is nonsingular at x_(o), where:    $\begin{matrix}    {{A(x)} = \begin{bmatrix}    {L_{g_{l}}L_{f}^{r_{i} - 1}{h_{l}(x)}} & \cdots & {L_{g_{m}}L_{f}^{r_{i} - 1}{h_{l}(x)}} \\    \vdots & ⋰ & \vdots \\    {L_{g_{l}}L_{f}^{r_{m} - 1}{h_{m}(x)}} & \cdots & {L_{g_{m}}L_{f}^{r_{m} - 1}{h_{m}(x)}}    \end{bmatrix}} & (3)    \end{matrix}$

The vector relative degree implies the multivariable notion of thesystem having no transfer function zeros, i.e., to ensure that no systemcharacteristic dynamics information is lost by observing the systemalong the output directions col{h_(i)(x);h₂(x); . . . ; h_(m)(x)}.

State Space Exact Linearization for Multi Input Systems

The state space exact linearization problem for multi-input systems canbe solved if an only if there exists a neighborhood U of x_(o) and mreal valued functions h₁(x), h₂(x), . . . , h_(m)(x) defined on U, suchthat the system (1) has vector relative degree {r₁, r₂, . . . , r_(m)}at x_(o) and ${{\sum\limits_{i = 1}^{m}r_{i}} = n},$with g(x_(o))=[g₁(x_(o)) g₂(x_(o)) . . . g_(m)(x_(o))] of rank m.

It remains to find the m output functions, h_(i)(x), satisfying theseconditions in order to determine the state variable transformation,given by the vectors:φ_(k) ^(i)(x)=L _(f) ^(k−1) h _(i)(x) ∀1≦k≦r _(i), 1≦i≦m  (4)then the diffeomorphism is constructed as: $\begin{matrix}{\Phi = \begin{bmatrix}{\phi_{1}^{l}(x)} \\\vdots \\{\phi_{r_{i}}^{l}(x)} \\\vdots \\{\phi_{1}^{m}(x)} \\\vdots \\{\phi_{r_{m}}^{m}(x)}\end{bmatrix}} & (5)\end{matrix}$Solving for the m Output Functions for LPD Systems. Two Inputs Example

It is directly demonstrated in this section how to find the outputfunctions for the LPD lateral dynamics of the aircraft model, withrudder and aileron inputs, and all state variables (sideslip, roll rate,yaw rate and bank angle) measured. Of course, other inputs and statevariables could be solved for as well. In this case, m=2 and the modelcan be written:{dot over (x)}=Ax+Bu  (6)with x∈R⁴ and u∈R². As will be appreciated by those of ordinary skill inthe art, variable A can represent an air vehicle dynamics matrix andvariable B can represent a control distribution matrix. Variable urepresents a vector of control, having variables corresponding to therudder and aileron, and x represents a system state vector (e.g.,x=[side slip, bank angle, roll rate, and yaw rate]). The vector relativedegree of the system is {r₁, r₂}={2,2} and the summation${\sum\limits_{i = 1}^{m}r_{i}} = n$is satisfied for r_(i)=2, m=2, and n=4. Evaluating the terms accordingto equation (2), yields for outputs y₁=C₁x and Y₂=C₂x:C₁B₁=0C₁B₂=0C₂B₂=0C₂B₂=0.C₁AB₁=1C₁AB₂=0C₂AB₁=0C₂AB₂=1  (7)where C_(i) is the i^(th) measurement direction. Note that the lowerfour equations of equation (7) satisfy the requirement that equation (3)be nonsingular. These can be rearranged in matrix form: $\begin{matrix}{\begin{bmatrix}\left. \leftarrow\left. C_{1}\rightarrow \right. \right. \\\left. \leftarrow\left. C_{2}\rightarrow \right. \right.\end{bmatrix}\left\lbrack \begin{matrix}\underset{\downarrow}{\overset{\uparrow}{B_{1}}} & \underset{\downarrow}{\overset{\uparrow}{B_{2}}} & {\underset{\downarrow}{\overset{\uparrow}{A}}\quad\underset{\downarrow}{\overset{\uparrow}{B_{1}}}} & {\left. {A\quad B_{2}} \right\rbrack = \begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1\end{bmatrix}}\end{matrix} \right.} & (8)\end{matrix}$which then allows solution for C₁ and C₂, following which, thetransformation matrix, or diffeomorphism, can be written according toequations (4) and (5), as $\begin{matrix}{\Phi = \begin{bmatrix}\left. \leftarrow\left. C_{1}\rightarrow \right. \right. \\\left. \leftarrow\left. C_{2}\rightarrow \right. \right. \\\left. \leftarrow\left. {C_{1}A}\rightarrow \right. \right. \\\left. \leftarrow\left. {C_{2}A}\rightarrow \right. \right.\end{bmatrix}} & (9)\end{matrix}$Feedback LTI'ing Control Law for a Two Input Lateral Aircraft DynamicsModel

The transformed z-space model is then determined, where z is the z-spacestate vector, Φ is the diffeomorphism, and x is the x-space statevector, from: $\begin{matrix}{z = {\Phi\quad x}} \\{\therefore} \\{\overset{.}{z} = {{\Phi\quad x} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}} \\{= {{\Phi\quad A\quad x} + {\Phi\quad B\quad u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}} \\{= {{\Phi\quad A\quad\Phi^{- 1}z} + {\Phi\quad B\quad u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}} \\{= {{A_{z}z} + {B_{z}\quad u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}}\end{matrix}\quad$which has the matrix form $\begin{matrix}{\begin{matrix}{\overset{.}{z} = {{\begin{bmatrix}{0\quad 0\quad 1\quad 0} \\{0\quad 0\quad 0\quad 1} \\\left. \leftarrow\left. {\alpha_{1}(z)}\rightarrow \right. \right. \\\left. \leftarrow\left. {\alpha_{2}(z)}\rightarrow \right. \right.\end{bmatrix}z} + {\begin{bmatrix}00 \\00 \\10 \\01\end{bmatrix}u} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}}}} \\{= {{\begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}z} + {\begin{bmatrix}00 \\00 \\10 \\01\end{bmatrix}v}}}\end{matrix}\quad} & (10)\end{matrix}$where the new dynamics and control distribution matrices are denotedA_(z) and B_(z), respectively. A new input is defined as follows tocancel the parameter dependent terms α₁(z) and α₂(z) and the summationterm which includes parameter rate of change terms (i.e., representingthe dependence of the coordinates transformation on the rate at whichthe parameters change), thus yielding the feedback LTI'ing control law,which together with the diffeomorphism of equation (9) has transformedthe LPD system (6) into an LTI system given by the last line of (10).$\begin{matrix}{v = {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {{\begin{bmatrix}{0\quad 0\quad 0\quad 0} \\{0\quad 0\quad 0\quad 0} \\\left. \leftarrow\left. {\alpha_{1}(z)}\rightarrow \right. \right. \\\left. \leftarrow\left. {\alpha_{2}(z)}\rightarrow \right. \right.\end{bmatrix}z} + {\sum\limits_{i = 1}^{q}{\frac{{\partial\Phi}\quad x}{\partial p_{i}}{\overset{.}{p}}_{i}}} + {B_{z}u}} \right\}}} & (11)\end{matrix}$

This system (10) is now ready for application of LTI feedback controldesign, while the LTI'ing control law of (11) ensures that the systemparameter dependence is accommodated in the physical control lawimplementation.

Solving for Gain Lookup Tables

The feedback LTI'ing control law (Eq. 11) for the two control input caseis written as: $\begin{matrix}{v = {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {{\begin{bmatrix}0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\\leftarrow & \alpha_{1} & (z) & \rightarrow \\\leftarrow & \alpha_{2} & (z) & \rightarrow\end{bmatrix}z} + {\sum\limits_{i = 1}^{q}\quad{\frac{\partial\Phi_{x}}{\partial p_{i}}{\overset{.}{p}}_{i}}} + {B_{z}u}} \right\}}} & (12)\end{matrix}$

For the full state feedback z-space control law of the form ν=−K_(z) zand ignoring the parameter rate of change terms for evaluating thelookup table gains (locally linear gains), the relationship betweenz-space locally linear gains and x-space locally linear gains is asfollows (where equivalently u=−K_(x)x): $\begin{matrix}\begin{matrix}{K_{x} = {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {\begin{bmatrix}0 & 0 & 0 & 0 \\0 & 0 & 0 & 0 \\\leftarrow & \alpha_{1} & (z) & \rightarrow \\\leftarrow & \alpha_{2} & (z) & \rightarrow\end{bmatrix} + {B_{z}K_{z}}} \right\}\Phi}} \\{= {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}\left\{ {A_{z} + {B_{z}K_{z}}} \right\}\Phi}}\end{matrix} & (13)\end{matrix}$

From this expression, it will be obvious to one of ordinary skill in theart that the lookup gains may be stored as either x-space lookup tables(K_(x)), or as z-space lookup tables (K_(z)). In the case of storing thez-space gains, it is also necessary to store the z-space matrices A_(z)and B_(z), as well as the diffeomorphism. This latter case amounts toreal time evaluation of z-space gains and then converting these to truephysical space gains as opposed to performing this transformationoff-line and simply storing the x-space gains in lookup tables.

Note that the full control law includes both the locally linear gainterm as well as the parameter rate of change term of equation (12).

Extending to Allow Design of Failure Detection Filter

U.S. Patent '119 shows the application to design a failure detectionfilter (FDF) for the single input case. The previous sections showed thegeneral multi-input solution of the feedback (“FBK”) LTI'zation problemresulting in a set of LTI equations of motion in z-space, with the samenumber of inputs as the original co-ordinates model. This section showsthe application of the ideas described in the '119 patent to the multiinput case, specifically, an example is given for the two input case. Aswill be appreciated by those skilled in the art, the more-than-two inputcase is a simple extension of the same form of equations.

With knowledge of the diffeomorsphism coefficients, it is now possibleto define the FDF in transformed coordinates, which will be the singlefixed-point design valid for the entire operating envelope of the system(e.g., aircraft). The failure detection filter is initially designed ata nominal operating point in the flight envelope, using the modeldescribed in physical coordinates, and taking advantage of the insightgained by working in these coordinates. This design is then transformedinto the z-space coordinates to determine the transformed space failuredetection filter which is then unchanged for all operating points in theflight envelope.

The z-space model is now, from equation (10), given by $\begin{matrix}{\overset{.}{z} = {{\begin{bmatrix}0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \\0 & 0 & 0 & 0 \\0 & 0 & 0 & 0\end{bmatrix}z} + {\begin{bmatrix}00 \\00 \\10 \\01\end{bmatrix}v}}} & (14)\end{matrix}$and the FDF appropriately designed for the relevant failure modes yieldsthe gain matrix H_(z) with the implemented system of the following form:{circumflex over(ż)}=[A _(z) −H _(z) ]{circumflex over (z)}+H _(z) Φx_(measured) +B _(z)ν  (15)or in physical vehicle coordinates,{circumflex over({dot over (x)})}=Φ ⁻¹ [A _(z) −H _(z) ]Φ{circumflexover (x)}+Φ ⁻¹ H _(z) Φx _(measured) +B _(x)ν  (16)where the subscript z refers to the transformed state space, and thesubscript x refers to physical coordinates state space. Also,x_(measured) refers to the measured state variables. For example, themeasured values of roll rate (p), yaw rate (r), bank angle (φ) andsideslip (β). Since the full state vector is measured, each of the statevariables is available in physical coordinates.

Returning to FIG. 2, the multi-input feedback LTI'zation control lawdesign example is given for the two input (aileron and rudder actuators)and two parameters (air density and dynamic pressure) case ofcontrolling the lateral dynamics of an aircraft, as shown in FIG. 1.Specifically, FIG. 2 show an example using reference bank angle (φref)and sideslip (βref) signals to control the aircraft 1. As will beappreciated by those of ordinary skill in the art, if a pilot wants theaircraft to fly right wing down ten degrees, he simply commands the bankangle of 10 degrees and the control law will cause the aircraft to flywith the right wing down 10 degrees. Likewise, if the pilot wants thenose of the aircraft to point 5 degrees to the left of the incomingairflow, this is achieved by commanding 5 degrees of sideslip, and thecontrol laws will cause the ailerons and rudder to move in such afashion as to deliver flight with the nose at 5 degrees to the incomingairflow.

As mentioned above, flight control laws are typically a plurality ofequations used to control flight in a predictable way. Flight controllaws are well known to those of ordinary skill in flight and vehiclecontrols and will not be described in greater detail herein. However,reference may be had to the text “AIRCRAFT DYNAMICS AND AUTOMATICCONTROL,” by McRuer, et al., Princeton University Press, 1973,incorporated herein by reference.

Known flight control laws for operating the aircraft rudder and aileronmay be simplified as:

Rudder Control Law (Equation 17a):δ  r = −G_(RdrBeta)(Beta − Beta  Ref) − G_(RdrRollRate)(RollRate) − G_(RdrYawRate)(YawRate) − G_(RdrRoll)(Roll − RollREf) − G_(RdrBetaIntegrator)∫(Beta − BetaRef)𝕕t − G_(RdrRollIntegrator)∫(Roll − RollRef)𝕕twhere δr represents commanded rudder deflection angle, “G” termsrepresent rudder control law gains, Beta represents measured sideslip,RollRate represents measured roll rate, Roll represents measured bankangle, RollRef represents a reference bank angle and YawRate representsthe measured yaw rate.

Aileron Control Law (Equation 17b):δ  a = −G_(AilBeta)(Beta − Beta  Ref) − G_(AilRollRate)(RollRate) − G_(AilYawRate)(YawRate) − G_(AilRoll)(Roll − RollREf) − G_(AilBetaIntegrator)∫(Beta − BetaRef)𝕕t − G_(AilRollIntegrator)∫(Roll − RollRef)𝕕twhere δa represents commanded aileron deflection angle, “G” termsrepresent aileron control law gains, Beta represents measured side slip(e.g., measured with a sensor), RollRate represents measured roll rate,Roll represents measured bank angle, RollRef represents a reference bankangle and YawRate represents the measured yaw rate.

The integrator terms shape the closed loop vehicle dynamics bycompensating for the steady state error which typically results withoutthese extra terms.

Including the parameter rate of change term, yields the final controllaw: $\begin{matrix}{\begin{bmatrix}{\delta\quad a} \\{\delta\quad r}\end{bmatrix} = {\begin{bmatrix}{\delta\quad a} \\{\delta\quad r}\end{bmatrix}_{{\overset{.}{p}}_{i} = 0} - {\left( {B_{z}^{\prime}B_{z}} \right)^{- 1}B_{z}^{\prime}{\sum\limits_{i = 1}^{q}\quad{\frac{\partial\Phi_{x}}{\partial p_{i}}{\overset{.}{p}}_{i}}}}}} & (18)\end{matrix}$

As discussed above, the “G” (gain) terms can be i) evaluated off-line,and stored in RAM look-up tables, and/or ii) evaluated in real-time, asdiscussed above with respect to Eq. 13.

As seen in FIG. 2, reference bank angle (φref) and reference side-slip(βref) signals 211 are compared (205) with sensor signals reflecting theaircraft's current bank angle (φ) and side slip (β), respectively. Thesevalues are input (or utilized by) into equations 17, along with sensorsignals representing the aircraft's actual roll-rate (p) and yaw-rate(r). The current dynamic air pressure (Q) and air density (ρ) areevaluated from sensor signals 201 and the corresponding gain values(e.g., “G_(i)”), implemented in one embodiment as RAM look-up tables asfunctions of aircraft dynamic pressure and air density 202, are appliedto the control laws 206. The appropriate gain value is determined byinterpolation between neighboring points in the lookup tables. Therequired number of gain look-up tables corresponds to the number ofstate variables plus any required integrals, multiplied by the number ofcontrol inputs (e.g., actuators). For example, the lateral axis of anaircraft has four (4) state variables (i.e., sideslip, bank angle, rollrate, and yaw rate) and two integrals (sideslip error and bank angleerror) for each actuator (i.e., rudder and aileron), for a total oftwelve (12) gain tables. Hence, in the lateral axis case, there aretwelve (12) corresponding look-up tables. If, for example, thelongitudinal axis were also considered, using an integral airspeed holdcontrol mode and actuating via the elevator, five more look-up tableswould be required. In the longitudinal axis case, there are four statevariables (i.e., angle of attack, pitch rate, pitch attitude, and trueairspeed) and one (1) integrator (i.e., the integral of airspeed minusairspeed reference). In another embodiment, the gain values can becalculated in real time, as discussed above with respect to Eq. 13.

The control gains (G_(i)) can be numerically evaluated in the controllaw design process in z-space, using Linear Quadratic Regulators (LQR)theory, a well defined and widely known LTI control design technique.Pole placement or any other known LTI technique could also be used. Aswill be appreciated by those of ordinary skill in the art, LQR theoryprovides a means of designing optimal control solutions for LTI systems.A quadratic cost function, which penalizes state variable excursions andactuator deflections in a weighted fashion, is solved for. The steadystate solution yields a set of gains and a specific full state feedbackcontrol law which defines how aircraft motion is fed back to deflectingthe control surfaces in order to maintain desired control at a constantoperating condition, i.e., constant parameter values.

At any specific operating condition, defined by ρ and Q, the controlgains are then transformed into x-space via equation (13), ignoring thedp/dt term for purposes of determining the gains, and stored in the RAMlook-up tables. The dp/dt term is included as per equation (18), withrelevant terms evaluated numerically, as discussed below. Alternatively,the values for the control gains can be determined in real time, everycomputational cycle (e.g., every 60 milliseconds or faster, depending onthe required processing speed for the specific aircraft application), bytwo (2) dimensional interpolation for the current air density anddynamic pressure that the aircraft is experiencing at any point in time,as discussed above with respect to Eq. 13. A second alternative is tofully determine the control commmands in z-space, and then transformthese to physical control commands using equation (12) and solving foru.

Parameter rate of change terms are evaluated numerically 203 for use inequation 18 (204). Parameter rate of change terms compensate for orcapture the varying rates of change of the operating conditions, asexperienced by the aircraft. For example, as an aircraft dives, the airdensity changes while the aircraft changes altitude. In order toaccommodate the effect of the changing density on the dynamic behaviorof the aircraft, the control system preferably accounts for the varyingair density.

An aircraft flying at high altitude and high true airspeed, typicallyexhibits much poorer damping of it's natural dynamics, than at lowaltitude and low airspeed. This effect varies as the speed and altitudevaries, and the rate of change of the altitude and airspeed alsoinfluence the dynamic behavior. In order to deliver similar closed loopbehavior even whilst, e.g., decelerating, the rate of change of dynamicpressure must be accounted for in the dp/dt terms of the control law.

Typically, parameters can be measured by means of a sensor, for example,dynamic pressure can be directly measured, but the rate of change of theparameter is not typically directly measured. In this case, the rate ofchange value is determined numerically through one of many known methodsof taking discrete derivatives, as will be appreciated by those skilledin the art. One example of such a numerical derivative evaluation is byevaluating the difference between a current measurement and the previousmeasurement of a parameter, and dividing by the time interval betweenthe measurements. This quotient will give a numerically evaluatedestimate of the rate of change of the parameter. These rate of changeparameter values are evaluated in real time. The other component of theparameter rate of change term, i.e., dφ/dp_(i) is also numericallyevaluated and can, however, be stored off-line in look-up tables orevaluated in real time.

Blocks 204 and 206 are combined (in summing junction 212) to yieldcomplete flight control commands for the rudder 103 and ailerons 101 (inblock 207). Signals are sent to the rudder and ailerons in 208,effecting control of the air vehicle dynamics (as shown in block 209).As mentioned above, sensors (210) feed back current roll rate, yaw rate,bank angle and slide-slip signals, and the above process is repeated,until the current measured bank angle and sideslip signals match therespective reference signals.

FIG. 3 is a flowchart depicting the software control carried out by theflight control computer 104. By way of illustration, a “decrabbing”maneuver is described in relation to FIG. 3. A decrabbing maneuver isexecuted when an aircraft experiences a crosswind while landing. Toperform the decrabbing maneuver, an aircraft on final approach tolanding faces into the crosswind, and then at a moment prior to landing,adjusts so that the nose of the aircraft is pointed down (i.e., isparallel to and along) the runway. For this decrabbing maneuver example,a 10 degree crosswind is imagined. To compensate for the crosswind, thecontrol system determines that in order to maintain a steady course(zero turn rate) whilst in a 10 degree sideslip condition, a 3 degreebank angle adjustment is also needed. Hence, in this example, thereference sideslip (βref) and bank angle (φref) are 10 degrees and 3degrees, respectively.

Referring to FIG. 3, in step S1, the reference sideslip (βref) and bankangle (φref) are input as reference signals. In step S2, the current airdensity (ρ) and dynamic pressure (Q) are input from sensors. In step S3,the rudder (δr) and aileron (δa) control laws are read from memory. Instep S4, the roll rate (p), yaw rate (r), bank angle (φ) and sideslip(β) signals are input from sensors.

In step S5, βref and φref are compared with β and φ signals fromsensors. In step S6, signals p and r, the comparison from step S5,sideslip and bank angle integrators, and control gains (Gi) are appliedto the rudder (δr) and aileron (δa) control laws. These control gains(G_(i)) are preferably off-line resolved to capture z-spacetransformations. In step S7, parameter rates of change terms aregenerated. Note that if the vehicle parameter rates of change are verysmall, these terms would be very small in magnitude and as such may beeliminated from the control law.

In step S8, the parameter rates of change generated in step S7 areapplied to the rudder (δr) and aileron (δa) control laws. In step S9, acontrol signal is output to the aileron 101 and rudder 103 actuators,effecting control of the aircraft. The control law integrators areincremented in step S10. In the de-crabbing maneuver example, the logicflow continues adjusting the rudder and aileron until β and φapproximate 10 and 3 degrees, respectively. In this manner, the controlsystem compensates for the 10-degree crosswind, by turning theaircraft's nose parallel to the runway just prior to landing. Ahigh-level control function may be implemented with the above-describedcontrol system to effect landing. For example, the high-level controlcould be a pilot or automatic control algorithm such as an autoland.

FIG. 4 is a block diagram showing the relationship between the varioussensors, actuators and the flight control computer 104. As can be seen,flight control computer 104 receives input from various sensors,including airspeed 105 a, altimeter 105 b, yaw rate 105 c, bank angle105 d, side slip 105 e, angle of attack 105 f, pitch rate 105 g, pitchattitude 105 h, roll rate 105 i and Nth sensor 105 n. The various sensorsignals are inserted into the appropriated flight control laws and theoutputs are actuator command signals, such as to the throttle 106 a,elevator 106 b, aileron 106 c, rudder 106 d, and Mth actuator 106 m.

Applying Feedback LTI'zation to Designing Control Laws

An example of the control law design techniques according to the presentinvention will now be described with respect to FIGS. 5 a-30.Essentially, the design process involves transforming the coordinatesfor the vehicle equations of motion into z-space. This step has beendetailed in equations 1-18, above. Known LTI control design techniquesare used as a framework for the control gains design process. Parametervalues at a few desired design points in the operating envelope areselected. LTI design techniques are applied to the physical LTI modelsat the (few) selected parameter values, to yield desired closed loopdynamics. These designs are transformed into the transformed coordinates(in z-space) to yield z-space gains that give desired closed loopbehavior for the controlled system. If more than one design point wasselected, these z-space gains are linearly interpolated over theoperating envelope, otherwise the gains are constant in z-space for thefull envelope. Finally, a discrete number of parameter values,corresponding to lookup table axes, is selected, and the reversetransformation applied to define the physical coordinates lookup gaintables for use as discussed above with respect to FIGS. 2 and 3, forexample.

By way of example, FIGS. 7-18 show lateral auto pilot gains in 3-D plotform for the lateral axis of the Perseus 004 aircraft aircraft over aflight envelope of sea level to 22 km altitude and 20 m/s to 46.95 m/sIAS). Each figure illustrates an auto pilot gain with respect to airdensity (kg/m³) and dynamic pressure (Pa). In particular, FIG. 7illustrates side slip to aileron feedback gain; FIG. 8 illustrates rollrate to aileron feedback gain; FIG. 9 illustrates yaw rate to aileronfeedback gain; FIG. 10 illustrates roll attitude to aileron feedbackgain; FIG. 11 illustrates side slip integrator to aileron feedback gain;FIG. 12 illustrates roll integrator to aileron feedback gain; FIG. 13illustrates side slip to rudder feedback gain; FIG. 14 illustrates rollrate to rudder feedback gain; FIG. 15 illustrates yaw rate to rudderfeedback gain; FIG. 16 illustrates roll attitude to rudder feedbackgain; FIG. 17 illustrates side slip integrator to rudder feedback gain;and FIG. 18 illustrates roll attitude integrator to rudder feedbackgain. Each of the represented gains is illustrated in physicalcoordinates.

FIGS. 19-30 are corresponding matrix numerical gain lookup tables forthe lateral axis of the Perseus 004 aircraft over a flight envelope ofsea level to 22 km altitude and 20 m/s to 46.95 m/s IAS. FIGS. 19-30provide the numerical data for FIGS. 7-18, respectively. The format foreach of FIGS. 19-30 is as follows: the first row is a dynamic pressurelookup parameter in Pa, the second row is a density lookup parameter inkg/m^3, and the remainder of rows are gain values.

Together, FIGS. 7-30 illustrate part of the design process fordetermining gains. In this example, optimal LQR designs are generated atfour discrete design points corresponding to the four corners of thecontrol design flight envelope (i.e., sea level to 22 km altitude and 20m/s to 46.95 m/s IAS). The four design points are thus low speed at highdensity; high speed at low density; low speed at low density; and highspeed at high density. Feedback LTI'zation is used to map these fourdesigns into smooth gain scheduling look-up tables at 110 points in adensity—dynamic pressure space. Each of the four (4) designs is done ata selected steady state flight condition (density, speed combination),in physical co-ordinates, since the designer readily understands these.The resulting gains (physical) at each design point are then transformedinto z-space, and by reversing this transformation (i.e., from z tophysical co-ordinates) at any parameter values, the physical coordinateslookup table values are populated. Typically, a selected matrix ofparameter values is defined, and the lookup tables are populated forthese parameter values. The process of populating the gain tables simplyexecutes equation 11, and does not require the designer to intervene ateach table lookup parameter value. This is a major reason for thesavings in design effort and time, namely that only a few design pointsare required, and then the full envelope is covered by appropriatetransformation using equation 11. Note that this results in physicalgains that can be used in real time in the control law. It is alsofeasible to perform real time reverse co-ordinates transformations fromthe z-space gains, at every time step, to determine the x-space gains inreal time. In this case, the lookup tables store z-space gains and notphysical x-space gains.

Root loci and step response data are used to evaluate performance androbustness during the design process. In particular, FIGS. 5 a-5 d showthe full envelope design results for a lateral auto pilot control systemdesign for the Perseus 004 aircraft. In particular, overlaid discretebode plots (i.e., FIGS. 5 c and 5 d) and step responses (i.e., FIGS. 5 aand 5 b) are shown for all combinations of air density and dynamicpressure in the gain tables covering the design envelope of sea level to22 km above sea level and 20 m/s to 46.95 m/s IAS. This illustrates howthe design can be used to achieve similar and well behaved closed loopperformance across the full envelope of operation, while requiring onlya very small number of design points—namely four design points in thisexample.

FIG. 6 illustrates the S-plane root loci for closed loop and open looplateral dynamics over the entire flight envelope for the Perseus 004aircraft, at the discrete density and dynamic pressure values in thelookup table. Open loop poles are circles, and closed loop poles arecrosses. Closed loop poles all lie inside the 45 degree sector from theorigin about the negative real axis, which is the design criterion forgood damping characteristics. By design, the closed loop modal frequencymagnitudes are not increased significantly over the open loop values.This reduces the danger of actuator saturation in normal envelopeoperation, as well as danger of delays due to too high closed loop modefrequencies for the sample period of 60 ms. The design goal of betterthan 70 percent damping of all modes is achieved, without significantlyaltering the modal frequencies.

This design example is particularly tailored between only four designpoints, namely one at each corner of the density/dynamic pressure space,which defines the flight control design envelope. At each of the fourdesign points, the well known LQR control design algorithm is used fordetermining the controller gains, as discussed above. The four pointdesigns are then transformed into a new set of coordinates, i.e., the socalled z-space via feedback LTI'zation routines, which are then furtherinvoked to determine the physical gains over the entire flight envelope,based on linear blending in z-space of the four point designs. The fourpoint designs yield four sets of gains in z-space, and these are simplylinearly interpolated between the four design points to provide linearlyblended gains in z-space. This technique achieves an approximatelylinear variation of closed loop bandwidth over the design envelope.

By transforming the single point design gains into z-space, where thedefinition of z-space forces the system to be LTI, the control gains atany other operating condition can be easily determined by simplyreversing the co-ordinates transformation process. This allows thesingle design point to be transformed into z-space and reversetransformed to an infinite number of operating conditions different fromthe design point. Including the parameter rate of change terms then alsoallows transition between design points without disturbing the closedloop behavior, in the sense that the closed loop characteristics remainconstant. It is this coordinates transformation step that allows a verysmall number of design points to cover the full operating envelope ofthe vehicle. Those of ordinary skill in the art will appreciate that the“few” design point case has very similar attributes to the single designpoint case.

Thus, what has been described is a control system (and method) tocontrol a dynamic device or system with multiple control inputs andmultiple parameter dependencies. An efficient method of control lawdesign has also been described for such multi-input, parameter dependentdynamic systems.

The individual components shown in outline or designated by blocks inthe drawings are all well known in the arts, and their specificconstruction and operation are not critical to the operation or bestmode for carrying out the invention.

While the present invention has been described with respect to what ispresently considered to be the preferred embodiments, it is to beunderstood that the invention is not limited to the disclosedembodiments. To the contrary, the invention is intended to cover variousmodifications and equivalent arrangements included within the spirit andscope of the appended claims. The scope of the following claims is to beaccorded the broadest interpretation so as to encompass all suchmodifications and equivalent structures and functions.

1. A method of determining the operational status of a multi-input,multi-parameter dependent control system for controlling a dynamicdevice, said method comprising the steps of: receiving a plurality ofcurrent status signals representing current statuses of the device and aplurality of control signals for effecting control of the device;transforming the current status signals and the control signals to amulti-input linear time invariant system; estimating an expectedbehavior of the device using the transformed current status signals andthe transformed control signals; and determining the operational statusof the control system by comparing the expected behavior of the deviceto the actual behavior of the device.
 2. The method of claim 1, furthercomprising the steps of: incrementally adjusting at least one parameterbased on the determined operational status of the control system; andupdating the determination of the operational status by repeating thereceiving, transforming, estimating, and determining steps.
 3. Themethod of claim 2, wherein characteristics of the dynamic device aredefinable by at least one control law, and wherein the step ofincrementally adjusting comprises the steps of: generating at least oneparameter rate of change term; and applying the at least one parameterrate of change term to the at least one control law.
 4. The method ofclaim 3, wherein the generating step is performed using linearinterpolation.
 5. The method of claim 3, wherein the generating step isperformed using extrapolation techniques.
 6. The method at claim 3,wherein the generating step is performed using curve fitting.
 7. Themethod of claim 2, wherein the dynamic device comprises a flightvehicle, and wherein the plurality of status signals includes at leastone of the group consisting of an airspeed signal, an altimeter signal,a yaw rate signal, a bank angle signal, a side slip signal, an angle ofattack signal, a pitch rate signal, a pitch attitude signal, and a rollrate signal.
 8. The method of claim 7, wherein the at least oneparameter corresponds to at least one of the group consisting of athrottle, an elevator, an aileron, and a rudder, and the step ofincrementally adjusting comprises using an actuator to physically adjustat least one of the group consisting of a throttle, an elevator, anaileron, and a rudder.
 9. The method of claim 7, wherein the flightvehicle is unmanned.
 10. The method of claim 2, wherein the steps ofincrementally adjusting and repeating are performed at predeterminedtime intervals.
 11. A multi-parameter dependent control system forcontrolling a dynamic device having multiple inputs, said systemcomprising: receiving means for receiving a plurality of current statesignals representing current states of the device and a plurality ofcontrol signals for effecting control of the device; transforming meansfor transforming the current state signals and the control signals to alinear time invariant system; estimating means for estimating anexpected behavior of the device using the transformed current statesignals and the transformed control signals and for generating estimatesignals corresponding to the expected behavior, and determining meansfor determining the operational status of the control system bycomparing the expected behavior of the device to the actual behavior ofthe device.
 12. The system of claim 11, further comprising: adjustingmeans for incrementally adjusting at least one parameter based on thedetermined operational status of the control system, wherein thedetermining means is further configured to update the determination ofthe operational status by repeating a use of the receiving means,transforming means, estimating means, and determining means when anincremental adjustment has been made.
 13. The system of claim 12,wherein characteristics of the dynamic device are definable by at leastone control law, and wherein the adjusting means is further configuredto: generate at least one parameter rate of change term; and apply theat least one parameter rate of change term to the at least one controllaw.
 14. The system of claim 13, wherein the adjusting means is furtherconfigured to generate the at least one parameter rate of change term byusing linear interpolation.
 15. The system of claim 13, wherein theadjusting means is further configured to generate the at least oneparameter rate of change term by using extrapolation techniques.
 16. Thesystem of claim 13, wherein the adjusting means is further configured togenerate the at least one parameter rate of change term by using curvefitting.
 17. The system of claim 12, wherein the dynamic devicecomprises a flight vehicle, and wherein the plurality of current statesignals includes at least one of the group consisting of an airspeedsignal, an altimeter signal, a yaw rate signal, a bank angle signal, aside slip signal, an angle of attack signal, a pitch rate signal, apitch attitude signal, and a roll rate signal.
 18. The system of claim17, wherein the at least one parameter corresponds to at least one ofthe group consisting of a throttle, an elevator, an aileron, and arudder, and wherein the adjusting means is further configured to use anactuator to physically adjust at least one of the group consisting of athrottle, an elevator, an aileron, and a rudder.
 19. The system of claim17, wherein the flight vehicle is unmanned.
 20. The system of claim 12,wherein the adjusting means is further configured to make an incrementaladjustment at predetermined time intervals.
 21. Apparatus for detectinga failure in a flight control device having multiple inputs andoperating in an environment having multiple parameters, comprising:processing structure for (i) receiving time-varying status signals fromthe flight control device, (ii) providing reference signalscorresponding to the flight control device, (iii) transforming both thestatus signals and the reference signals to a linear time invariantcoordinate system, (iv) calculating flight control device estimatesignals based on the transformed status signals and the transformedreference signals, (v) transforming the calculated estimate signal in aphysical coordinate system, and (vi) detecting an error in the flightcontrol device when a difference between the transformed estimatesignals and the status signals exceed a predetermined threshold.
 22. Theapparatus of claim 21, wherein the time-varying status signals includeat least one of the group consisting of an airspeed signal, an altimetersignal, a yaw rate signal, a bank angle signal, a side slip signal, anangle of attack signal, a pitch rate signal, a pitch attitude signal,and a roll rate signal.
 23. The apparatus of claim 21, wherein theprocessing structure is further configured to calculate flight controlestimate signals using at least one control law.
 24. The apparatus ofclaim 21, wherein the processing structure is further configured toupdate the calculated flight control device estimate signals atpredetermined time intervals.
 25. The apparatus of claim 21, wherein theflight control device is unmanned.